Stan is a great tool with many state-of-the-art algorithms for Bayesian statistics. Infer.NET features great and mature inference algorithms as well. With Infer.NET, you can easily build models that can be naturally expressed as factor graphs. You can't
do that in Stan. The two frameworks are not direct competitors in my opinion. They have slightly different audiences.

In Stan you can't have discrete random variables. Authors argue that these can be taken care using marginalization but for new users and beginners in Bayesian statistics this is not that easy.

To my knowledge, Infer.NET is the only mature probabilistic programming framework that supports true online learning. That is, you can use the posterior as prior in the next iterations. There are some limitations and technical details but I managed to get
that working in my models. This is very useful when data cannot fit in memory. Stan and PyMC3 don't support online learning.